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An Intuitive Introduction to Fractional and Rough Volatilities mathematics Review An Intuitive Introduction to Fractional and Rough Volatilities Elisa Alòs 1,* and Jorge A. León 2 1 Department d’Economia i Empresa, Universitat Pompeu Fabra, and Barcelona GSE c/Ramon Trias Fargas, 25-27, 08005 Barcelona, Spain 2 Departamento de Control Automático, CINVESTAV-IPN, Apartado Postal 14-740, 07000 México City, Mexico; [email protected] * Correspondence: [email protected] Abstract: Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments. Keywords: derivative operator in the Malliavin calculus sense; fractional Brownian motion; future average volatility; Hull and White formula; Itô’s formula; Skorohod integral; stochastic volatility models; implied volatility; skews and smiles; rough volatility Citation: Alòs, E.; León, J.A. An MSC: 60G22; 91G20 Intuitive Introduction to Fractional and Rough Volatilities. Mathematics 2021, 9, 994. https://doi.org/ 10.3390/math9090994 1. Introduction Academic Editor: Marianito Rodrigo It is well-known that the classical Black-Scholes model [1] describes the current market behavior when it is assumed that the volatility process s is a constant. However, despite its Received: 1 April 2021 simplicity, empirical observations show that some important features of option prices are Accepted: 23 April 2021 not represented by this model. Hence, the Black-Scholes model (11) has to be extended to Published: 28 April 2021 the case where the volatility s is a stochastic process. A simple method to achieve this is to allow the volatility s to be a process independent of the noise governing the stock prices Publisher’s Note: MDPI stays neutral (see Renault and Touzi [2], Stein and Stein [3], and Scott [4], amongst others). Under this with regard to jurisdictional claims in model, some features, such as the smile, are analyzed using the Hull and White formula [5] published maps and institutional affil- (see (15) below), which can be obtained via the Itô’s formula and states that the price of the iations. European option is given by a conditional expectation of the Black-Scholes option pricing formula where the constant volatility is changed by the future average volatility s Z T 1 2 Copyright: © 2021 by the authors. t ss ds, t [0, T]. (1) 7! T t t 2 Licensee MDPI, Basel, Switzerland. − This article is an open access article where T is the maturity time. distributed under the terms and The study of the financial data showed that correlation exists between the volatility s conditions of the Creative Commons and the price process (see, for instance, Bates [6], Heston [7], and Johnson and Shanno [8]). Attribution (CC BY) license (https:// Consequently, we need to consider extensions of model (11). In order to fix ideas, we now creativecommons.org/licenses/by/ 4.0/). Mathematics 2021, 9, 994. https://doi.org/10.3390/math9090994 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 994 2 of 22 suppose that the asset price follows the dynamics of the stochastic differential equation (in the Itô’s sense) q dS = rS dt + s S (rdW + 1 r2dB ), t [0, T], (2) t t t t t − t 2 where W and B are two independent Brownian motions and s is a process adapted to the filtration generated by W. So, in order to analyze the properties of the market represented by the model for stock prices (2), we have to identify a Hull and White type formula for this model. However, in this case, we cannot apply the classical Itô’s calculus techniques since the future average volatility (1) is not an adapted process to the filtration generated by W and B. So, it is necessary to deal with a stochastic integral that allows us to integrate processes that are not adapted to the underlying filtration. As such, Malliavin calculus becomes a useful tool for the study of models with stochastic volatility. In particular, this theory does not require the volatility to be either a diffusion or a Markov process. Thus, it is now possible to work with fractional volatilities, which satisfy long and short dependence, as conducted by Alòs et al. [9] in 2007. In this paper, we briefly describe the analytical approach used in the literature to deal with the problem of establishing Hull and White type formulas for different financial markets with stochastic volatilities (see Alòs [10] for the original idea), and introduce the techniques of Malliavin calculus to provide methods for analytical and numerical approaches to examine option pricing problems. It is also well-known that the stochastic volatility models with diffusion as a volatility process capture the important features of the implied volatility as the smile (or skew) and term structure (see Barndorff-Nielsen and Shephard [11,12], Bates [6], Fouque et al. [13], and Renault and Touzi [2], amongst others). The implied volatility is the process that fits the Black-Scholes price formula with the market price of an observed European call. The Hull and White type formula provides a useful tool for calculating the derivative of the implied volatility with respect to log-strike, which depends on the derivative of s in the Malliavin calculus sense, asexplained in this paper. Thus, the Hull and White formula becomes an important technique for studying the at-the-money short-term behavior skew slopes, even for fractional volatilities (see Alòs et al. [9]). The main purpose of this paper is to provide a brief introduction of the tool needed to obtain and understand some results of fractional volatility models. We explain how these models improve the description of some empirical findings of the implied volatility using the long and short memory of the underlying driving fractional process s. The paper is organized as follows: The fractional Brownian motion is introduced in Section2. In Section3, we describe the framework that we use in this paper, namely the basic tools of the Malliavin calculus that we need to establish the results ofinthe paper. In Section4 , we consider several volatility models. The implied volatility is reviewed in Sections4 and5. The analytical study of the Hull and White formula and its consequences on the implied volatility are explained in Section6. Finally, in Section7, we consider mixed fractional Bergomi models, whose volatility s combines long and short memories. 2. Fractional Brownian Motion It is well-known that Itô’s calculus [14] for Brownian motion W = W : t [0, T] has f t 2 g a wide range of applications in the fields of human knowledge via stochastic differential equations. This calculus is based on two important tools: Itô’s integral and Itô’s formula, which allow us to deal with stochastic processes. The Itô’s integral is not, in general, a Riemann–Stieltjes integral due to W having non-bounded variation paths; Itô’s formula is a type of fundamental theorem of calculus. The construction of these two tools uses either the martingale property or the independence of increases in W. However, a natural restriction for Itô’s calculus is that the integrands have to be adapted to the filtration (information) W generated by W. So, by the Doob–Meyer decomposition theorem, the classical Itô’s F calculus is extended to semimartingales as integrators. Among the applications of classical Itô’s calculus is the Black-Scholes formula in mathematical finance [1]. Mathematics 2021, 9, 994 3 of 22 Despite the number of applications of Itô’s calculus for Brownian motion, we cannot consider phenomena that exhibit long-range dependence [15]. That is, the covariance of the increases in the involved process on intervals is non-zero and decays slowly as a negative power of the distance between the intervals. As examples, the long dependence appears in stock price changes (see Greene and Fielitz [16]), hydrology (see Mandelbrot and Wallis [17]), rainfall (see Mandelbrot [18], and Mandelbrot and Wallis [19]), amongst others). In volatility modeling, Comte and Renault [20] observed that the long-maturity behavior of the implied volatility can be explained by long-memory volatilities, pioneering the use of the fractional Brownian motion in volatility modeling. Some other processes are observed to satisfy short memory. That is, the correlation between increments is negative and has a fast decay as a function of the distance between intervals. Even though these short-range properties are less studied, short-memory pro- cesses have been proved to be of interest in the modeling of volatility process in finance (see Alòs et al. [9] and Gatheral et al. [21]). Hence, we need to consider processes satisfying long- and short-range dependence, as does fractional Brownian motion (fBm). However, fBm is not a semimartingale in general (see Roger [22]). Therefore, it is necessary to develop techniques of stochastic calculus for fBm that cannot be obtained from classical Itô’s calculus.
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